Problem: You have found the following ages (in years) of 6 sloths. The sloths are randomly selected from the 50 sloths at your local zoo: $ 4,\enspace 19,\enspace 14,\enspace 16,\enspace 6,\enspace 12$ Based on your sample, what is the average age of the sloths? What is the variance? You may round your answers to the nearest tenth.
Solution: Because we only have data for a small sample of the 50 sloths, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{60.84} + {51.84} + {4.84} + {17.64} + {33.64} + {0.04}} {{6 - 1}} $ $ {s^2} = \dfrac{{168.84}}{{5}} = {33.77\text{ years}^2} $ We can estimate that the average sloth at the zoo is 11.8 years old. There is a variance of 33.77 years $^2$.